Are there online store order processing systems (OMS) that work through an external API?

Question

CRM

Help with advice, please.
There is a self-written LP that sells only one position (one product). For such a simple online store, I don't want to reinvent the wheel in the form of OMS (Order Management System) plus CRM (Customer Management System).
I would like to use a third-party system, into which I can push an order from my page through the API and keep the entire order history and the entire customer base in this external system.
The main thing is that it changes the status of orders, logs activity with orders and customers, maintains a history of orders, groups orders by couriers, builds simple reports.
Was available via the web. Ideally, it had mobile applications and sent notifications of new orders.
And it was not necessary to install it at home, but was available as a web service (SAAS).
So far, nothing like it has been found.
Thanks in advance for the recommendations, I really hope that such systems exist.

Answer 1

My warehouse seems to be doing this, I also saw www.retailcrm.ru recently, sort of like

Answer 2

There is also ru.orderino.com
Just do not forget that the Saas also have disadvantages. When you start growing, you will want much more flexible integration.